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Theorem cotr3 14378
 Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)
Hypotheses
Ref Expression
cotr3.a 𝐴 = dom 𝑅
cotr3.b 𝐵 = (𝐴𝐶)
cotr3.c 𝐶 = ran 𝑅
Assertion
Ref Expression
cotr3 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem cotr3
StepHypRef Expression
1 cotr3.a . . 3 𝐴 = dom 𝑅
21eqimss2i 3952 . 2 dom 𝑅𝐴
3 cotr3.b . . . 4 𝐵 = (𝐴𝐶)
4 cotr3.c . . . . 5 𝐶 = ran 𝑅
51, 4ineq12i 4116 . . . 4 (𝐴𝐶) = (dom 𝑅 ∩ ran 𝑅)
63, 5eqtri 2782 . . 3 𝐵 = (dom 𝑅 ∩ ran 𝑅)
76eqimss2i 3952 . 2 (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
84eqimss2i 3952 . 2 ran 𝑅𝐶
92, 7, 8cotr2 14377 1 ((𝑅𝑅) ⊆ 𝑅 ↔ ∀𝑥𝐴𝑦𝐵𝑧𝐶 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539  ∀wral 3071   ∩ cin 3858   ⊆ wss 3859   class class class wbr 5033  dom cdm 5525  ran crn 5526   ∘ ccom 5529 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ral 3076  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536 This theorem is referenced by: (None)
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